Nakai-Moishezon criteria and the toric Thomas-Yau conjecture

TL;DR

This paper proves a version of the Thomas-Yau conjecture for Lagrangian sections in certain Calabi-Yau fibrations, linking stability conditions to Hamiltonian isotopy to special Lagrangians via mirror symmetry and Hermitian Yang-Mills connections.

Contribution

It establishes a new stability criterion for Lagrangian sections in Calabi-Yau fibrations, connecting mirror symmetry, the Nakai-Moishezon criterion, and the Thomas-Yau conjecture.

Findings

Hamiltonian isotopy to special Lagrangians characterized by stability conditions

Bridgeland stability implies Hamiltonian isotopy to special Lagrangians on certain mirrors

Unstable Lagrangians relate to solutions of the special Lagrangian equation with phase conditions

Abstract

We consider a class of Lagrangian sections $L$ contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: $L$ is Hamiltonian isotopic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. We use the SYZ transform, the toric gamma theorem, and toric homological mirror symmetry in order to reduce the statement to one about supercritical deformed Hermitian Yang-Mills connections, known as the Nakai-Moishezon criterion. As an application, we prove that, on the mirror of a toric weak del Pezzo surface, if $L$ defines a Bridgeland stable object in the Fukaya-Seidel category in a natural sense, then it is Hamiltonian isotopic to a special Lagrangian section in the class. The converse also holds for the mirror of the projective plane blown-up at one or two points, and always holds assuming a conjecture of Arcara and Miles. When $L$ is Bridgeland unstable, we obtain a morphism from $L$ to a weak solution of the special Lagrangian equation with phase angle satisfying a minimality condition. These results are consistent with general conjectures due to Joyce. We discuss some generalisations, including a weaker analogue of our main result for general projective toric manifolds, and a similar obstruction, related to Lagrangian multi-sections, in a special case.